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Geometric rescaling algorithms for submodular function minimization

Dadush, Daniel, Végh, László A. ORCID: 0000-0003-1152-200X and Zambelli, Giacomo (2021) Geometric rescaling algorithms for submodular function minimization. Mathematics of Operations Research, 46 (3). 1081 - 1108. ISSN 0364-765X

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Identification Number: 10.1287/moor.2020.1064

Abstract

We present a new class of polynomial-time algorithms for submodular function minimization (SFM) as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige–Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. First, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound O((n4 · EO + n5)log(nL)). Second, we exhibit a general combinatorial black box approach to turn εL-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige–Wolfe algorithm. Combined with the geometric rescaling technique, the black box approach provides an O((n5 · EO + n6)log2n) algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee et al., yielding a simplified variant of their O(n3log2n · EO + n4logO(1)n) algorithm.

Item Type: Article
Official URL: https://pubsonline.informs.org/journal/moor
Additional Information: © 2021 INFORMS
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 13 Feb 2020 16:09
Last Modified: 20 Sep 2021 03:16
URI: http://eprints.lse.ac.uk/id/eprint/103368

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